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<Paper uid="P93-1027">
  <Title>ON THE DECIDABILITY OF FUNCTIONAL UNCERTAINTY*</Title>
  <Section position="4" start_page="0" end_page="202" type="metho">
    <SectionTitle>
2 '\]?he Method
</SectionTitle>
    <Paragraph position="0"> We will first briefly describe the main part of solving the standard feature terms and then turn to their extension with functional uncertainty.</Paragraph>
    <Paragraph position="1"> Consider a clause C/ = xplyl A xpzy2 (from now on we will refer to pure conjunctive formulae as clauses). A standard method for solving feature terms would rewrite C/ in order to achieve a solved form. This rewriting depends on the paths Pl and Pz. If Pl equals Pz, we know that yl and Y2 must be equal.</Paragraph>
    <Paragraph position="2"> This implies that C/ is equivalent to xplyx Ayl -- Yz. If  p~ is a prefix of p2 and hence P2 = P~P~, we can transform C/ equivalently into the formulae xplyi A YlP'Y2. The reverse case is treated in a similar'fashion. If neither prefix or equality holds between the paths, there is nothing to be done. By and large, clauses where this holds for every x and every pair of different constraints xp~y and xp2z are the solved forms in Smolka \[9\], which are consistent.</Paragraph>
    <Paragraph position="3"> If we consider a clause of the form C/ = zL~y~ A zL2y~, then we again have to check the relation between ys and y~. But now there is in general no unique relation determined by C/, since this depends on which paths p~ and P2 we choose out of L~ and L~. Hence, we have to guess the relation between pl and p~ before we can calculate the relation between yl and y~. However, there is a problem with the original syntax, namely that it does not allow one to express any relation between the chosen paths (in a later section we will compare our algorithm to the one of Kaplan/Maxwell, thus showing where exactly the problem occurs in their syntax). Therefore, we extend the syntax by introducing so-called path variables (written c~, fl, a',...), which are interpreted as feature paths (we will call the other variables first order variables). Hence, if we use the modified subterm relation xo~y and a restriction constraint o~ ~ L, a constraint xLy can equivalently be expressed as xay A a ~ L (4 new). The interpretation of xay is done in two steps. Given a valuation V~, of the path variables as feature paths, a constraint =c~y in C/ is substituted by xV~,(cQy. This constraint is then interpreted using the valuation for the first order variables in the way such constraints are usually interpreted. null By using this extended (two-sorted) syntax we are now able to reason about the relations between different path variables. In doing so, we introduce additional constraints c~ - fl (equality), o~ ~ fl (prefix) and c~ fl fl (divergence). Divergence holds if neither equality nor prefix holds. Now we can describe a normal form equivalent to the solved clauses in Smolka's work, which we will call pre-solved clauses. A clause C/ is pre-solved iff for each pair of different constraint xayl and x~y2 in ~b there is a constraint a I\] ~ in C/. We call this clauses pre-solved, since such clauses are not necessarily consistent. It may happen, that the divergence constraints together with the restrictions of form a ~ L are inconsistent (e.g. think of the clause a~ f+ A ~ ~ ff+ A (~ fl fl). But pre-solved clauses have the property, that if we find a valuation for the path variables, then the clause is consistent.</Paragraph>
    <Paragraph position="4"> Our algorithm first transforms a clause into a set of pre-solved clauses, which is (seen as a disjunction) equivalent to the initial clause. In a second phase the pre-solved clauses are checked for consistency with respect to the path variables. In this paper we will concentrate on the first phase, since it is the more difficult one.</Paragraph>
    <Paragraph position="5"> Before looking at the technical part we will illustrate the first phase. For the rest of the paper we will write clauses as sets of atomic constraints. Now consider the clause 7 = {xay, al ~ L1, xflz, fl~ L2}.</Paragraph>
    <Paragraph position="6"> The first step is to guess the relation between the path variables c~ and ft. Therefore, 7 can be expressed equivalently by the set of clauses</Paragraph>
    <Paragraph position="8"> The clause 71 is pre-solved. For the others we have to evaluate the relation between a and \]Y, which is done as follows. For 72 we substitute/~ by ot and z by y, which yields {y &amp;quot;--z, xay, o~E L1, aEL2}.</Paragraph>
    <Paragraph position="9"> We keep only the equality constraint for the first order variables, since we are only interested in their valuation. Combining {4 ~ L1, a ~ L2} to {4 ~ (L1 f')L2)} then will give us the equivalent pre~solved clause For 73 we know that the variable/3 can be split into two parts, one of them covered by 4. We can use concatenation of path variables to express this, i.e. we can replace fl by the term c~.fl', where ~' is new. Thus we get the clause 7~ - {xc~y, a~ L1, yfl' z, c~.fl'~L2}, The only thing that we have to do additionally in order to achieve a pre-solved clause is to resolve the constraint a./~ ~ ~ L2. To do this we have to guess a so-called decomposition P, S of L2 with P.S C_ L2 such that a ~ P and \]~' ~ S. In general, there can be an infinite number of decompositions (think of the possible decompositions of the language f'g). But as we use regular languages, there is a finite set of regular decompositions covering all possibilities. Finally, reducing {c~ ~ L~, ~ ~ P} to {~ ~ (L1 n P)} will yield a pre-solved clause.</Paragraph>
    <Paragraph position="10"> Note that the evaluation of the prefix relation in 73 has the additional effect of introducing a new constraint y~z. This implies that there again may be some path variables the relation of which is unknown.</Paragraph>
    <Paragraph position="11"> Hence, after reducing the terms of form a --&amp;quot; \]~ or ~ fl we may have to repeat the non-deterministic choice of relations between path variables. In the end, the only remaining constraints between path variables will be of the form a fl ft.</Paragraph>
    <Paragraph position="12"> We have to consider some additional point, namely that the rules we present will (naturally) loop in some cases. Roughly speaking, one can say that this always occurs if a cycle in the graph coincides with a cycle in the regular language. To see this let us vary the above example and let 7 now be the clause {xax, c~ ~ f, xflz, fl ~ f'g}. Then a possible looping derivation could be  1. adda4\]~: {4 4 fl, xax, a~f, xflz, fl~f*g} 2. split fl into a-f~': 3. decompose c~-/~ I~ f'g: {=~, ~f, ~f~'~, a~f*, Z'~f*g}</Paragraph>
  </Section>
  <Section position="5" start_page="202" end_page="202" type="metho">
    <SectionTitle>
4. join a-restrictions:
</SectionTitle>
    <Paragraph position="0"> {=~z, ~I, ~/~'z, ~'~y*g} However, we will proof that the rule system is quasi-terminating, which means that the rule system may cycle, but produces only finitely many different clauses (see \[4\]). This means that checking for cyclic derivations will give us an effective algorithm.</Paragraph>
    <Paragraph position="1"> Quasi-termination is achieved by the following measures: first we will guarantee that the rules do not introduce additional variables; second we restrict concatenation to length 2; and third we will show that the rules system produces only finitely many regular languages. In order to show that our rewrite system is complete, we also have to show that every solution can be found in a pre-solved clause.</Paragraph>
  </Section>
  <Section position="6" start_page="202" end_page="204" type="metho">
    <SectionTitle>
3 Preliminaries
</SectionTitle>
    <Paragraph position="0"> Our signature consists of a set of sorts S (A, B,...), first order variables X (z,y,...), path variables 7 9 (a,/3,...) and features Jr (f, g,...). We will assume a finite set of features and infinite sets of variables and sorts. A path is a finite string of features. A path u is a prefix of a path v (written u ~ v) if there is a non-empty path w such that v = uw. Note that is neither symmetric nor reflexive. Two paths u, v diverge (written u n v) if there are features f, g with f ~ g and possibly empty paths w, wl, w2 such that u = wfw~ A v = wgw2. Clearly, n is a symmetric relation.</Paragraph>
    <Paragraph position="1"> Proposition 3.1 Given two paths u and v, then exactly one of the relations u = v, u .~ v, u ~- v oru II v holds.</Paragraph>
    <Paragraph position="2"> A path term (p, q .... ) is either a path variable a or a concatenation of path variables a.fl. We will allow complex path terms only in divergence constraints and not in prefix or equality constraints. Hence, the set of atomic constraints is given by</Paragraph>
    <Paragraph position="4"> We exclude empty paths in subterm agreement since xey is equivalent to x - y. Therefore, we require fl&amp;quot;...'fn E ~r+ and L C_ jr+.</Paragraph>
    <Paragraph position="5"> A clause is a finite set of atomic constraint denoting their conjunction. We will say that a path term a.fl is contained (or used) in some clause C/ if C/ contains either a constraint a-fl ~ L or a constraint a.fl ti q) Constraints of the form p~ L, p fl q, a :~ fl and c~ - fl will be called path constraints.</Paragraph>
    <Paragraph position="6"> An interpretation Z is a standard first order structure, where every feature f ~ ~ is interpreted as a binary, functional relation F z and where sort symbols We will not differentiate between p fl q and q ~ p.</Paragraph>
    <Paragraph position="7"> are interpreted as unary, disjoint predicates (hence A zOBz= 0 for A 5PS B). A valuation is a pair (Vx, VT~), where Vx is a standard first order valuation of the variables in X and Vv is a function V~v : P ---+ ~'+. We define V~,(a.fl) to be VT,(a)V~,(13), The validity of an atomic constraint in an interpretation 2&amp;quot; under a valuation (Vx, V~,) is defined as follows:</Paragraph>
    <Paragraph position="9"> for aC {u,k,--&amp;quot; }, where p is the path fl&amp;quot;...'f, and F/z are the interpretations of fi in Z.</Paragraph>
    <Paragraph position="10"> For a set ~ C X we define =PS to be the following relation on first order valuation: Vx =~ V/~ iff W e ~ : Vx(~) = V/~(x).</Paragraph>
    <Paragraph position="11"> Similarly, we define =~ with 7r C 79 for path valuations. Let 0 C_ XU79 be a set of variables. For a given interpretation 7: we say that a valuation (Vx, V~) is a O-solution of a clause C/ in 2&amp;quot; if there is a valuation (V~, V~) in 2&amp;quot; such that Vx =a'ne V~:, Vp =~,no V~ and (V~:, V~) ~z C/. The set of all 0-solutions of C/ in 2: is denoted by \[C/\]~. We will call X-solutions just solutions and write \[C/\]z instead of \[C/\],~.</Paragraph>
    <Paragraph position="12"> For checking satisfiability we will use transformation rules. A rule R is O-sound C/ --*n 7</Paragraph>
    <Section position="1" start_page="202" end_page="204" type="sub_section">
      <SectionTitle>
4.1 A Set of Rules
</SectionTitle>
      <Paragraph position="0"> Recall that we have switched from the original syntax to a (two-sorted) syntax by translating constraints zLy into {zay, ~ ~ L}, where a is new. The result of the translation constitutes a special class of clauses, namely the class of prime clauses, which will be defined below. Hence, it suffices to show decidability of consistency of prime clauses. They are the input clauses for the first phase.</Paragraph>
      <Paragraph position="1"> Let C/ be some clause and z, y be different variables. We say that C/ binds y t0 z if z - y E C/ and y occurs only once in C/. Here it is important that we consider equations as directed, i.e. we assume that z -&amp;quot; y is different from y - x. We say that C/ elimi- null nates y if C/ binds y to some variable x. A clause is called basic if 1. x - y appears in C/ iff C/ eliminates y, 2. For every path variable a used in C/ there is at most one constraint zc~y E C/.</Paragraph>
      <Paragraph position="3"> A basic clause C/ is called prime if C/ does not contain an atomic constraint of the form p fl q, c~ -~/3 or ot - null /3. Every clause C/ in the original Kaplan/Maxwell syntax can be translated into a prime clause 7 such that C/ is consistent iff 9' is consistent.</Paragraph>
      <Paragraph position="4"> Now let's turn to the output clauses of the first step. A basic clause is said to be pre-soived if the following holds: 1. Ax 6 C/ and Bz 6 C/5 implies A - B.</Paragraph>
      <Paragraph position="5"> 2. c~ d L 6 C/ and a d L' 6 C/ implies L = L*.</Paragraph>
      <Paragraph position="6"> Furthermore, a d O is not in C/.</Paragraph>
      <Paragraph position="7"> 3. a-/3, c~ -/3 or a ~/3 are not contained in C/. 4. afl/36C/iffa~/3, x(~y6C/andz/3z6C/.</Paragraph>
      <Paragraph position="8"> Lemma 4.1 A pre-soived clause C/ is consistent iff  there is a path valuation V~, with VT~ ~ Cp, where Cp is the set of path constraints in ~.</Paragraph>
      <Paragraph position="9"> Now let's turn to the rule system. As we have explained informally, the first rule adds nondeterministiely relational constraints between path variables. In one step we will add the relations between one fixed variable a and all other path variables/3 which are used under the same node x as a. Furthermore, we will consider only the constraints - /3, c~ fl /3 and a ~ /3 and not additionally the constraint a 9/3.</Paragraph>
      <Paragraph position="10"> For better readability we will use pseudo-code for describing this rule (using the usual don't care/don't know distinction for non-determinism):  For each x/3z 6 C/ with c~ #/3 and c~ fl/3 ~ C/ add a 6~/3 with 5Z 6 {-, 4~, fl} (don't know) &amp;quot;don't care non-determinism&amp;quot; means that one is free to choose an arbitrary alternative at this choose point, whereas &amp;quot;don't know&amp;quot; means that one has to consider every alternative in parallel (i.e. for every alternative of the don't care non-determinism a clause C/ is equivalent to the set of all don't know alternatives that can be generated by applying the rule to C/). Note that the order of rule application is another example for don't care non-determinism in our rule system.</Paragraph>
      <Paragraph position="11"> Although we have restricted the relations 6~ to {-, :(, u}, this rule is globally preserving since we have non-deterministically chosen zay. To see this let C/ be a clause, 27 be an interpretation and (Vx, VT~) be a valuation in 27 with (Vx, V~) ~z C/. To find an instance of (PathRel) such that (Vx, V~,) ~z 7 where 3' is the result of applying this instance, we choose xay 6 C/ with V~(a) is prefix minimal in {v~@ 1~/3z ~ C/}.</Paragraph>
      <Paragraph position="12"> Then for each x/3z 6 C/ with a #/3 and ~ fi /3 ~ C/ we add a 6~ /3 where Vp(a) o~ V~(/3) holds. Note that 5 0 equals ~ will not occur since we have chosen a path variable a whose interpretation is prefix minimal. Therefore, the restriction 6~ 6 {-, k, fi} is satisfied.</Paragraph>
      <Paragraph position="13"> We have defined (PathRel) in a very special way.</Paragraph>
      <Paragraph position="14"> The reason for this is that only by using this special definition we can maintain the condition that concatenation of path variables is restricted to binary concatenation. E.g. assume that we would have added both /31 &amp;quot;~ O~ and a :C/ /32 to a clause 7. Then first splitting up the variable a into/31 .a' and then 132 into a./3~ will result in a substitution of/32 in 7 by/31&amp;quot;a&amp;quot;/3~. By the definition of (PathRel) we have ensured that this does not occur.</Paragraph>
      <Paragraph position="15"> The second non-deterministic rule is used in the decomposition of regular languages. For decomposition we have the following rules:</Paragraph>
      <Paragraph position="17"> where P, S, L C F + and A is a finite set of reg. languages with L, P, S 6 A. L must contain a word w with \[w\[ &gt; 1.</Paragraph>
      <Paragraph position="18"> The clash rule is needed since we require regular languages not to contain the empty path. The remaining rules are listed in Figure 1.</Paragraph>
      <Paragraph position="19"> We use A in (LangDecA) as a global restriction, i.e. for every A we get an different rule (LangDecA) (and hence a different rule system 7~A). This is done because the rule system is quasi-terminating. By restricting (LangDeca) we can guarantee that only finitely many regular languages are produced.</Paragraph>
      <Paragraph position="20">  For (LangDec^) to be globally preserving we need to find a suitable pair P, S in A for every possible valuation of (~ and \]3. Therefore, we require A to</Paragraph>
      <Paragraph position="22"> We will call A closed under decomposition if it satisfies this condition. Additionally we have to ensure that L E A for every L that is contained in some clause C/. We will call such a set A C-closed. Surely, we will not find a finite A that is closed under decomposition and C-closed for arbitrary C/. But the next lemma states some weaker condition that suffices. We say that 7 is a (C/,TiA)-derivative if 7 is derivable from C by using only rules from 7~h. If R^ is clear from the context, we will just say that 7 is a  1. If A is C-closed and closed under intersection, then A is 7-closed for all (C, T~h)-derivaLives 7.</Paragraph>
      <Paragraph position="23"> 2. For every prime clause C there is a finite A such that A is C-closed and closed under intersection  and decomposition.</Paragraph>
      <Paragraph position="24"> The proof of this lemma (containing the construction of the set A) can be found in the appendix.</Paragraph>
    </Section>
    <Section position="2" start_page="204" end_page="204" type="sub_section">
      <SectionTitle>
4.2 Completeness and Quasi-Termination
</SectionTitle>
      <Paragraph position="0"> The rule system serves for an algorithm to transform a prime clause into an equivalent set of pre-solved clauses. The rules are applied in arbitrary order until a pre-solved clause has been derived. If one of the non-deterministic rules is applied, a clause is substituted by a whole set of clauses, one for each of the don't know alternatives. Since the rule system is quasi-terminating, we may encounter cycles during the application of the rules. In this case we skip the corresponding alternative, since every pre-solved clause that can be produced via a cyclic derivation can also be produced via a derivation that does not contain a cycle.</Paragraph>
      <Paragraph position="1"> Theorem 4.3 Let C/ be a prime clause. If A is Cclosed, closed under intersection and decomposition, then \[\[C\] z = U.y~ \[\[7\] z for every interpretation Z, where C/b is the set of pre-solved (C, T~^)-derivatives. The set (9 is finite and effectively computable.</Paragraph>
      <Paragraph position="2"> To prove this theorem we have to show that the rule system is sound and complete. Sound means, that we do not add new solutions during the processing, whereas complete means that we find all solutions in the set of pre-solved derivatives.</Paragraph>
      <Paragraph position="3"> For the completeness it normally suffices to show that (1) every rule preserves (or globally preserves) the initial solutions and (2) the pre-solved clauses are exactly the T~h-irreducible clause (i.e. if a clause is not pre-solved, then one rule applies). But in our case this is not sufficient as the rule system is quasiterminating. A prime clause C/ may have a solution Vx which is a solution of all (C, T~A)-derivatives in some cyclic derivation, but can not be found in any pre-solved (C/, T~h)-derivative. We have to show that this cannot happen. Since this part of the proof is unusual, we will explain the main idea (see the appendix for a more detailed outline of the proofs).</Paragraph>
      <Paragraph position="4"> Let C/ be some (consistent) prime clause and let Vx E ~C/\]z for some Z. Then there exists a path valuation Vp such that (Vx, V~) ~z C/. We will find a pre-solved C-derivative that has Vx as a solution by imposing an additional control that depends on V~,.</Paragraph>
      <Paragraph position="5"> This control will guarantee (1) finiteness of derivations, (2) that each derivation ends with a pre-solved clause, (3) the initial solution is a solution of every clause that is derivable under this control. Since the (Pre) rule does not preserve the initial path valuation V~, (recall that the variable fl is substituted by the term a.~), we have to change the path valuation V~, every time (Pre) is applied. It is important to notice that this control is only used for proof purposes and not part of the algorithm. For the algorithm it suffices to encounter all pre-solved e-derivatives.</Paragraph>
      <Paragraph position="6"> To understand this control, we will compare derivations in our syntax to derivations in standard feature logic. Recall that we have a two-level interpretation. A constraint xay is valid under Vx and V~ if xV~(c~)y is valid under Vx. Hence, for each clause C/ and each valuation Vx, Vp with C valid under Vx and Vp there is a clause Cv~ in standard feature logic syntax (not containing functional uncertainty) such that C/v~ is valid under Vx. E.g. for the clause {xax, a~f, xflz, fl~f*g} and a path valuation V~, with VT,(a) = f and V~,(j3) = g the clause Cv~, is {xfx, xgy}. The control we have mentioned requires (by and large) that only those rewrite rules will be applied, that are compatible to the clause Cv~ and thus preserve Vx. If one of the rules (Eq) or (Pre) is applied, we also have to rewrite Cv~. Taking the above example, we are only allowed to add ali fl to C (using (PathRel)), since ev~ is already in pre-solved form.</Paragraph>
      <Paragraph position="7"> Now let's vary the example and let Vp be a path valuation with V~,(a) = f and V~,(f~) = Hg. Then we have to add a ~ /3 in the first step, since this relation holds between a and ft. The next step is to apply (Pre) on a :~ /3. Here we have to rewrite both C/ and Cv~. Hence, the new clauses C/1 and evv are {xax, a~f, x/3z, a./3~ f*g} and {x f x, x fgy} respectively. Note that the constraint xffgy has been reduced to x fg y by the application of (Pre).</Paragraph>
      <Paragraph position="8"> Since infinite derivations must infinitely often use (Pre), this control guarantees that we find a pre-solved clause that has Vx as a solution.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="204" end_page="205" type="metho">
    <SectionTitle>
5 The Second Phase
</SectionTitle>
    <Paragraph position="0"> In the second phase we have to check consistency of pre-solved clauses. As we have mentioned, a pre-solved clause is consistent if we find some appropriate path valuation. This means that we have to check the consistency of divergence constraints of the form al fi a2 together with path restrictions  al ~ L1 and a2 ~ L2. A constraint al ti a2 is va|id under some valuation V~, if there are (possibly empty) words w, wl, w2 and features'f ~ g such that V~,(al) = WfWl and V~,(c~2) = wgw2. This definition could directly be used for a rewrite rule that solves a single divergence constraint, which gives us {al fi ct2} UC/ f#g, ~,~1 2new where C/' = C/\[al ~--/?.a~,a2 ~/3.a~\]. By the application of this rule we will get constraints of the form j3.a~ ~ L1 and fl.a~ ~ L2. Decomposing these restriction constraints and joining the corresponding path restrictions for ~ and ~,~ will result in {fl~ (Pl nP2), ~i ~ (S~:*ns,), ,~ (g~'*MS2)} with PI.S~ C L~ and P2.S2 C_ L~, which completes the consistency check.</Paragraph>
    <Paragraph position="1"> Additionally, one has to consider the effects of introducing the path terms/~.a~. The main part of this task is to resolve constraints of the form fl.tr~ li tr. There are two possibilities: Either a has also f~ as an prefix, in which case we have to add fl ~ a; or fl is not a prefix of c~, which means that we have to add c~ fl ft. After doing this, the introduced prefix constraints have to be evaluated using (Pre). (In the appendix we present a solution which is more appropriate for proofing termination).</Paragraph>
  </Section>
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